More precisely, we study the existence of nonnegative nontrivial solutions of the problem −[M(kukp1,p)]p−1∆pu=λH(u−a)uq+h(x)us in Ω, u= 0 on∂Ω Mathematics Subject Classification

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF SOLUTIONS TO NONLOCAL ELLIPTIC EQUATIONS WITH DISCONTINUOUS TERMS

FRANCISCO JULIO S. A. CORR ˆEA, RUBIA G. NASCIMENTO

Abstract. In this article, we study the existence of nonnegative solutions for the elliptic partial differential equation

−[M(kuk^p_1,p)]^1,p∆pu=f(x, u) in Ω, u= 0 on∂Ω,

where Ω⊂R^Nis a bounded smooth domain,f: Ω×R⁺→Ris a discontinuous nonlinear function.

1. Introduction

This article concerns the existence of solution to the elliptic problem

−[M(kuk^p_1,p)]^1,p∆pu=f(x, u) in Ω,

u= 0 on∂Ω, (1.1)

where Ω⊂R^N is a bounded smooth domain, f : Ω×R⁺ →Ris a discontinuous function,M :R⁺→R,R⁺= [0,∞), ∆p is the p-Laplacian

∆_pu= div(|∇u|^p−2∇u), p >1, andk · k1,pis the usual norm

kuk^p_1,p= Z

Ω

|∇u|^p in the Sobolev spaceW₀^1,p(Ω).

The interest of the mathematicians on the so called nonlocal problems like (1.1) (nonlocal because of the presence of the termM(kuk^p_1,p), has increased because they represent a variety of relevant physical and engineering situations and requires a nontrivial apparatus to solve them.

More precisely, we study the existence of nonnegative nontrivial solutions of the problem

−[M(kuk^p_1,p)]^p−1∆_pu=λH(u−a)u^q+h(x)u^s in Ω,

u= 0 on∂Ω, (1.2)

2000Mathematics Subject Classification. 35A15, 35J40, 34A36.

Key words and phrases. Variational methods; elliptic problem; discontinuous nonlinearity.

c

2012 Texas State University - San Marcos.

Submitted October 12, 2011. Published February 7, 2012.

F.J.S.A.C. was supported by grants 620150/2008-4 and 303080/2009-4 from CNPq/Brazil.

R.G.N. was supported by grant 505407/2008-6 from CNPq/Brazil.

1

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where M :R⁺ →Ris a continuous function, 1< q+ 1< p < s+ 1< p^∗ = _N−P^pN , a > 0 and λ > 0 are real parameters, h : Ω → (0,∞) is a positive measurable function,h∈L^∞(Ω) and H is the Heaviside function

H(t) =

(0 ift≤0, 1 ift >0.

We assume the following conditions:

(H1) There exist m1, t1>0 such thatM(t)≥m1if 0≤t≤t1; (H2) There exist m2, t2>0 such that 0< M(t)≤m2 ift≥t2; (H3) lim_t→∞[M(t^p)]^p−1t^(p−1)−q = +∞;

(H4) M is non-increasing andM(t)>0 for allt >0.

Problems involving discontinuous nonlinearity appears in several physical situations. Among these, we may cite electrical phenomena, plasma physics, free boundary value problems, etc. The reader may consult Ambrosetti-Calahorrano- Dobarro [5], Ambrosetti-Turner [6], Arcoya-Calahorrano [7], Arcoya-Diaz-Tello [8], Badialle [9], [10], and the references therein. Some physical problems are related to discontinuous surface

Γa(u) ={x∈Ω;u(x) =a}

which causes difficulties in analyzing this kind of problems.

WhenM ≡1, (1.2) becomes a local problem, and has been widely studied. In particular, Alves-Bertone [1] and Alves-Bertone-Goncalves [2] use variants of the Mountain Pass Theorem (for locally Lipschitz functionals), the Ekeland Variational Principle, and the Subdifferential Calculus. On the other hand, after the work by Alves, Correa and Matofu [3] several papers appeared dealing with nonlocal problems with variational techniques; see for example [13, 14, 15, 16, 17, 20, 21, 24, 25].

This article maybe the the first study of a nonlocal problem with variational techniques for a non-differentiable functional. We consider the non-differentiable functional

Iλ,a(u) =1

pMc(kuk^p_1,p)−λψ(u)− 1 s+ 1

Z

Ω

h(x)(u⁺)^s+1, defined onW₀^1,p(Ω), where

Mc(t) = Z t

0

[M(s)]^p−1ds, ψ(u) = Z

Ω

F(u), F(u) =

Z u 0

f(t)dt, f(t) =H(t−a)(t⁺)^q, t⁺= max{0, t}.

By a solution to (1.2) we mean a functionu∈W₀^1,p(Ω)∩W^p,^q+1^q (Ω) satisfying

−[M(kuk^p_1,p)]^p−1∆_pu(x)−h(x)u(x)^s∈λ[f(u(x)), f(u(x))] a.e. in Ω, (1.3) wheref(t) =H(t−a)(t⁺)^q is nondecreasing,

f(t) = lim

δ→0⁺f(t+δ), f(t) = lim

δ→0⁺f(t−δ).

Let us consider the level set

Γ_a(u) ={x∈Ω;u(x) =a}.

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Note that if|Γa(u)|= 0, then usatisfies

−[M(kuk^p_1,p)]^p−1∆_pu(x) =λH(u(x)−a)u(x)^q+h(x)u(x)^s a.e. in Ω. (1.4) Clearly, a solution in the sense of (1.3) is also a solution in the sense of (1.4). The main result of this work is as follows.

Theorem 1.1. Suppose that M satisfies assumptions(H1)–(H4) andh∈L^∞(Ω).

Then there are λ^∗ > 0 and a^∗ > 0 such that for λ ∈ (0, λ^∗) and a ∈ (0, a^∗), Problem (1.2) possesses at least two nontrivial and nonnegative solutions u1 and u2 satisfying

(i) |Γa(ui)|= 0,i= 1,2;

(ii) Iλ,a(u2)<0< Iλ,a(u1);

(iii) |{x∈Ω;ui(x)> a}|>0,i= 1,2.

2. Abstract Framework

In this section we establish some basic results on critical point theory for locally Lipschitz functionals, developed by Chang [11] based on Convex Analysis and on the Subdifferential Calculus by Clarke [12].

Definition 2.1. LetX be a Banach space. We say that the functionalI:X →R is locally Lipschitz (I ∈Lip_loc(X,R)) if, givenu∈X, there exist a neighborhood V ≡Vu⊂X, u∈V and a constantk≡kV >0 such that

|I(v2)−I(v₁)| ≤kkv2−v₁k, v₁, v₂∈V.

Definition 2.2. The directional derivative of the locally Lipschitz functionalI : X →Ratu∈X in the direction v∈X is defined by

I⁰(u;v) = lim sup

h→0,λ→0⁺

I(u+h+λv)−I(u+h)

λ .

We may prove thatI⁰(u;v) is subadictive and positively homogeneous; that is, I⁰(u;v1+v2)≤I⁰(u;v1) +I⁰(u;v2)

and

I⁰(u;λv) =λI⁰(u;v) for allu, v1, v2∈X andλ >0.

Using these properties, it follows that

|I⁰(u, v₁)−I⁰(u, v₂)| ≤K|v1−v₂|, K≡K_u>0.

Consequently,I⁰(u,·) is continuous and, because it is also convex, we may consider its subdifferential atz∈X which is given by

∂I⁰(u;z) ={µ∈X^∗;I⁰(u;v)≥I⁰(u;z) +hµ, v−zi, v∈X},

whereX^∗ is the topological dual ofX andh·,·iis the duality pairing betweenX^∗ andX.

Definition 2.3. The generalized gradient of I∈Lip_loc(X,R) at u∈X is defined as being the set

∂I(u) ={µ∈X^∗;I⁰(u;v)≥ hµ, vi, for allv∈X}.

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SinceI⁰(u; 0) = 0, it follows that∂I(u) =∂I⁰(u; 0). Furthermore, for allv∈X, we have

I⁰(u;v) = max{hµ, vi;v∈∂I(u)}.

An important property of the generalized gradient is as follows: ifu∈X, then∂I(u) is a convex, nonempty and weak^∗−compact. In particular, there isω∈∂I(u) such that

m(u) = min{kωk_∗;ω∈∂I(u)}.

The reader may find more properties on this subject in [12] and [18]. We note that

∂I(u) ={I⁰(u)} whenI∈C¹(X,R).

Definition 2.4. A sequence (u_n) ⊂ X is a Palais-Smale sequence at the level c ((P S)_c), if

I(u_n)→c, m(u_n)→0.

Definition 2.5. We say that the functionalI ∈Lip_loc(X,R) satisfies the Palais- Smale condition at the levelc, if any (P S)_csequence possesses a strongly convergent subsequence.

The proof of our main result rests heavily on the following version of the Moun- tain Pass Theorem for Lip_loc functionals whose proof may be found in Chang [11].

Its proof uses an appropriate version of the Deformation Lemma whose proof is found in [18].

We say thatu0 ∈ X is a critical point of I if 0∈ ∂I(u0). Clearly, every local minimum (maximum) point is a critical point.

Theorem 2.6. LetI∈Lip_loc(X,R) be a functional such thatI(0) = 0 and suppose that:

(i) There are constantsη >0 andρ >0 such thatI(u)> η, forkuk=ρ, u∈X; (ii) There ise∈X, withkek> ρ, such thatI(e)<0.

If, in addition,Isatisfies the Palais-Smale condition at the level c= inf

γ∈Γ max

t∈[0,1]I(γ(t)), where

Γ ={γ∈C([0,1], X), γ(0) = 0 andγ(1) =e}, thenc >0 is a critical value of I.

3. Preliminary Results

In this section we establish some results for proving the main result of this article.

Lemma 3.1. There is λ^∗>0 such that the functionalIλ,a satisfies the geometric conditions(i) and(ii)of Mountain Pass Theorem 2.6, for alla >0

Proof. (i) Let us consider u∈ W₀^1,p(Ω) such that 0 <kuk^p_1,p =r < t1. Then, by (H1),

Iλ,a(u)≥ m^p−1₁

p kuk^p_1,p−λ Z

Ω

Z u 0

H(t−a)(t⁺)^qdt dx− 1 s+ 1

Z

Ω

h(x)(u⁺)^s+1 Noticing thatH≤1 andu⁺≤ |u|, we obtain

Iλ,a(u)≥m^p−1₁

p kuk^p_1,p− λ

q+ 1|u|^q+1_q+1− 1 s+ 1

Z

Ω

h(x)(u⁺)^s+1.

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From the Sobolev immersions and from the fact thath∈L^∞(Ω), we obtain I_λ,a(u)≥m^p−1₁

p r^p− C₁λ

q+ 1r^q+1−C₂r^s+1. Choosingr >0 sufficiently small, there exists

λ^∗= m^p−1₁ (q+ 1) 4pC1r^(q+1)−p

such thatIλ,a(u)≥η >0 forkuk1,p=r, for allλ∈(0, λ^∗), and for someη >0.

(ii) Letϕbe a function inC₀^∞(Ω), ϕ >0 in Ω. Hence, fort >0 withktϕk^p_1,p> t2, it follows from (H2) and recalling thatH ≥0, we have

Iλ,a(tϕ)≤ 1

pm^p−1₂ t^pkϕk^p_1,p− 1 s+ 1

Z

Ω

h(x)(u⁺)^s+1+Ce

and because 1< q+ 1 < p < s+ 1, we haveIλ,a(tϕ)→ −∞ as t → +∞. This

completes the proof.

Remark 3.2. Using Lemma 3.1 we may infer, from the Mountain Pass Theorem for Lip_locfunctionals, the existence of a sequence (un)⊂W₀^1,p(Ω) such thatIλ,a(un)→ cand m(un)→0.

The proof of the following lemma can be found in [11].

Lemma 3.3. If u∈W₀^1,p(Ω) andω∈∂ψ(u), then

ω(x)∈[f(u(x)), f(u(x))] a.e. inΩ.

In what follows, for the functionalIλ,a, we will use notation Iλ,a(u) =φ(u)−λψ(u)−J(u), where

φ(u) = 1

pMc(kuk^p_1,p), ψ(u) = Z

Ω

F(u), J(u) = 1 s+ 1

Z

Ω

h(x)(u⁺)^s+1. Lemma 3.4. The functional Iλ,a satisfies the Palais-Smale condition.

Proof. Let (u_n)⊂W₀^1,p(Ω) be a sequence satisfyingI_λ,a(u_n)→c andm(u_n)→0.

For the rest of this article, consider (ωn)⊂(W₀^1,p(Ω))^∗ be such thatm(un) = kωnk_∗ and

ωn =φ⁰(un)−λρn−J⁰(un) (3.1) with (ρ_n)⊂∂ψ(u_n).

Claim 3.5. The sequence(u_n)⊂W₀^1,p(Ω)is bounded.

Indeed, from (3.1),

hωn+λρn, uni= [M(kunk^p_1,p)]^p−1kunk^p_1,p− Z

Ω

h(x)(u⁺_n)^sun

and so

Iλ,a(un)− 1

s+ 1hωn+λρn, uni

=1

pMc(kunk^p_1,p)−λ Z

Ω

F(un)− 1 s+ 1

Z

Ω

h(x)(u⁺_n)^s+1

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− 1

s+ 1[M(kunk^p_1,p)]^p−1kunk^p_1,p+ 1 s+ 1

Z

Ω

h(x)(u⁺_n)^sun. Writingu_n =u⁺_n −u⁻_n, we obtain

1 s+ 1

Z

Ω

h(x)(u⁺_n)^su_n= 1 s+ 1

Z

Ω

h(x)(u⁺_n)^s+1, which implies

I_λ,a(u_n)− 1

s+ 1hωn+λρ_n, u_ni

= 1

pMc(ku_nk^p_1,p)−λ Z

Ω

F(u_n)− 1 s+ 1

Z

Ω

h(x)(u⁺_n)^s+1

− 1

s+ 1[M(kunk^p_1,p)]^p−1kunk^p_1,p+ 1 s+ 1

Z

Ω

h(x)(u⁺_n)^s+1, from which it follows that

I_λ,a(u_n)− 1

s+ 1hωn+λρ_n, u_ni

= 1

pMc(ku_nk^p_1,p)− 1

s+ 1[M(ku_nk^p_1,p)]^p−1ku_nk^p_1,p−λ Z

Ω

F(u_n).

(3.2)

Since (un) is a (P S)csequence, there exists a constantC2>0 such that|Iλ,a(un)| ≤ C2for alln∈N. Because

1

s+ 1[−hωn, uni]≤ 1

s+ 1|hωn, uni| ≤ 1

s+ 1kωnk∗kunk1,p≤C3kunk1,p, by (3.2), we obtain

I_λ,a(u_n)− 1

s+ 1hω_n+λρ_n, u_ni ≤C₂+ 1

s+ 1|hω_n, u_ni|+ 1

s+ 1|hλρ_n, u_ni|

≤C₂+C₃ku_nk_1,p+ λ

s+ 1|hρ_n, u_ni|.

Since (ρ_n)⊂∂ψ(u_n),hρn, vi ≤ψ⁰(u_n, v) for allv∈W₀^1,p(Ω).

Using arguments found in [11] and [12], we can show that hρn, uni ≤ψ⁰(un;un)

≤ Z

{u_n<0}

f(u_n)u_n+ Z

{u_n>0}

f(u_n)u_n

≤ Z

{un>0}

|un|^q+1≤ Z

Ω

|un|^q+1

≤C4kunk^q+1_1,p . and

Iλ,a(un)− 1

s+ 1l∠ωn+λρn, uni ≤C2+C3kunk1,p+C5kunk^q+1_1,p . (3.3) From inequalities (3.2)-(3.3), it follows that

1

pMc(kunk^p_1,p)− 1

s+ 1[M(kunk^p_1,p)]^p−1kunk^p_1,p

≤C2+C3kunk1,p+C5kunk^q+1_1,p +λ Z

Ω

Z u_n 0

(t⁺)^qdt dx.

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Using the Sobolev immersions, 1

p

Z kunk^p_1,p 0

[M(s)]^p−1ds− 1

s+ 1[M(ku_nk^p_1,p)]^p−1ku_nk^p_1,p

≤C2+C3kunk1,p+C7kunk^q+1_1,p .

(3.4)

From the continuity of M^p−1 on [0,kunk^p_1,p] and in view of Mean Value Theorem for integrals, there existsξn, 0< ξn<kunk^p_1,p, such that

1 p

Z kunk^p_1,p 0

[M(s)]^p−1ds= [M(ξn)]^p−1kunk^p_1,p. SinceM is a nonincreasing function, we obtain

[M(ξn)]^p−1≥[M(kunk^p_1,p)]^p−1 (3.5) and

1

p[M(kunk^p_1,p)]^p−1kunk^p_1,p− 1

s+ 1[M(kunk^p_1,p)]^p−1kunk^p_1,p

≤C2+C3kunk1,p+C7kunk^q+1_1,p , from which

1 p− 1

s+ 1

[M(kunk^p_1,p)]^p−1kunk^(p−1)−q_1,p ≤ C2

kunk^q+1_1,p + C3

kunk^q_1,p +C7. From (H3), we conclude that (u_n) is bounded. Showing the claim.

Since{un}is (P.S) sequence, using standard arguments, we can assume, without loss of generality, thatun≥0 for allx∈Ω.

As (un) is bounded and using the reflexivity ofW₀^1,p(Ω) there areu1∈W₀^1,p(Ω) andϑ∈Rsuch that, up to a subsequence,

kunk^p_1,p→ϑ^p, u_n * u₁ in W₀^1,p(Ω).

Consequently,u1≥0.

Let us now show that un → u1 in W₀^1,p(Ω). From the continuity of M and kunk^p_1,p→ ϑ^p, we obtain M(kunk^p_1,p)→ M(ϑ^p) and because M(ϑ^p) >0, there is K >0 such that

M(ku_nk^p_1,p)≥K >0 fornlarge enough.

Using the well known Simon inequality (see [23]), we obtain K^p−1Cp

Z

Ω

|∇un− ∇u1|^p

≤[M(kunk^p)]^p−1 Z

Ω

h|∇un|^p−2∇un− |∇u1|^p−2∇u1,∇un− ∇u1i and so

K^p−1Cp

Z

Ω

|∇un− ∇u1|^p

≤[M(kunk^p_1,p)]^p−1kunk^p_1,p−[M(kunk^p_1,p)]^p−1 Z

Ω

|∇un|^p−2∇un∇u1

−[M(kunk^p_1,p)]^p−1 Z

Ω

|∇u1|^p−2∇u1∇un+ [M(kunk^p_1,p)]^p−1ku1k^p_1,p.

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Noticing that

[M(ku_nk^p_1,p)]^p−1ku₁k^p_1,p−[M(ku_nk^p_1,p)]^p−1 Z

Ω

|∇u₁|^p−2∇u₁∇u_n=o_n(1), we obtain

K^p−1Cp

Z

Ω

|∇un− ∇u1|^p

Ω

|∇un|^p−2∇un∇u1+on(1).

We point out that

Z

Ω

h(x)u^s+1_n → Z

Ω

h(x)u^s+1₁ Z

Ω

h(x)u^s₁u1→ Z

Ω

h(x)u^s+1₁

|hρn, u1i − hρn, uni|=|hρn, uni − hρn, u1i|

=|hρn, un−u1i|

≤ kρ_nk_∗|u_n−u₁|_p→0,

From (3.1) and boundedness of{un}, it follows that (ρn) is bounded in (W₀^1,p(Ω))^∗, and sinceun →u1 inL^α(Ω), 1≤α < p^∗ we obtain

λ(hρ_n, u₁i − hρ_n, u_ni) =o_n(1).

We may write K^p−1Cp

Z

Ω

|∇un− ∇u1|^p

Ω

|∇un|^p−2∇un∇u1

− Z

Ω

h(x)u^s+1_n + Z

Ω

h(x)u^s_nu₁−λhρn, u_ni+λhρn, u₁i+o_n(1)

=hω_n, u_ni−iω_n, u₁i=o_n(1).

Hence,un →u1inW₀^1,p(Ω) which shows thatIλ,asatisfies the (P S)ccondition.

4. Proof of the Theorem 1.1

Part I: Multiplicity of solutions. Using Lemmas 3.1 and 3.4, from the Mountain Pass Theorem for Lip_loc functionals, it follows thatu₁ is a critical point ofI_λ,a at the levelc; i.e.,

Iλ,a(u1) =c >0 (4.1) which implies thatu₁6≡0.

Since {un} is (P.S) sequence, there are {ωn} ⊂∂Iλ,a(un) and{ρn} ⊂∂Ψ(un) verifyingkωnk_∗→0 and

hωn, φi= [M(kunk^p_1,p)]^p−1 Z

Ω

|∇un|^p−2∇un∇ϕ− Z

Ω

h(x)u^s_n−λ Z

Ω

ρ_nϕ (4.2) where, by Lemma 3.3,

ρ_n∈[f(u_n(x)), f(u_n(x))] a.e in Ω. (4.3)

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The boundedness of {un} combined with (4.3) implies in particular that {ρn} is bounded inL^q+1^q (Ω). Thus, there isρ₀∈L^q+1^q (Ω) such that, up to a subsequence ρn* ρ0 inL^q+1^q (Ω), or equivalently

Z

Ω

ρ_nϕ→ Z

Ω

ρ₀ϕ, ∀ϕ∈L^q+1(Ω) (4.4)

By [4, Lemma 3.3],ρ0(x)∈[f(u1(x)), f(u1(x))] in Ω.

Lettingn→+∞in (4.2), and using (4.4), we obtain the identity [M(ku1k^p_1,p)]^p−1

Z

Ω

|∇u1|^p−2∇u1∇ϕ− Z

Ω

h(x)u^s₁ϕ=λ Z

Ω

ρ0ϕ.

Showing thatu1 is a weak solution of the problem

−[M(ku1k^p_1,p)]^p−1∆_pu₁−hu^s₁=λρ₀ in Ω u1>0 in Ω.

By elliptic regularity, onceρ0∈L^q+1^q (Ω), it follows thatu1∈W^p,^q+1^q (Ω) and

−[M(ku1k^p_1,p)]^p−1∆pu1(x)−h(x)u^s₁(x) =λρ0(x) a.e. in Ω which implies

−[M(ku1k^p_1,p)]^p−1∆pu1(x)−h(x)u^s₁(x)∈λ[f(u1(x)), f(u1(x))] a.e. in Ω.

This shows thatu₁ is a solution of (1.2).

Proof of (i). Let us show that|Γ_a(u₁)|= 0, where Γ_a(u₁) ={x∈Ω;u₁(x) =a}.

Let us suppose, by contradiction, that|Γ_a(u₁)|>0. From the Morrey-Stampacchia Theorem [22],−∆_pu₁(x) = 0 a.e. in Γ_a(u₁), and so

−[M(ku1k^p_1,p)]^p−1∆pu1(x) = 0 a.e. in Γa(u1). (4.5) Sinceu₁is a critical point, it follows that

−[M(ku1k^p_1,p)]^p−1∆pu1(x)−h(x)u^s₁(x)∈λ[f(u1(x)), f(u1(x))] a.e. in Ω.

From (4.5), we obtain

−h(x)u^s₁(x)∈λ[f(u1(x)), f(u1(x))] a.e. in Ω.

As 0≤H(u₁−a)(u⁺₁)^q ≤(u⁺₁)^q, it follows from the definition off(u₁(x)), f(u₁(x)) and from the fact thatu₁≥0, that

0≤f(u1(x))≤f(u1(x))≤(u1)^q.

Thus,−h(x)u^s₁(x)∈[0, λa^q] which is impossible. Hence|Γ_a(u₁)|= 0.

Second Solution (Ekeland Variational Principle). By Lemma 3.1, we obtain I_λ,a(u)≥η for 0<kuk1,p=ρand so I_λ,a(u) is bounded from below onB_r and so there is inf_B

rI_λ,a(u).

Claim 4.1. There isa^∗ >0 such that for a∈(0, a^∗) we haveinf_B_r₍₀₎Iλ,a(u)<0.

Indeed, let us define the auxiliary function

ϕτ(x) =







τ a |x| ≤¹₂, 2τ a(1− |x|) ¹₂ ≤ |x| ≤1,

0 |x| ≥1.

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whereτ >(q+ 2)^1/(q+1). We point out thatϕτ ∈W₀^1,p(Ω) and kϕτk^p_1,p=

Z

Ω

|∇ϕτ|^p= Z

{|x|≤1}

|∇ϕτ|^p= (2τ a)^p Z

{|x|≤1}

|∇|x|)|^p.

Using the change of variables x=ωτ, withω∈S^N−1 impliesdx=τ^N⁻¹ds(ω)dτ. Then we obtain|x|=τ. Hence, _∂x^∂τ

i = ^x_τⁱ which implies|∇r|= 1. In this way, kϕτk^p_1,p≤(2τ a)^pαN,

whereαN is the volume of the unit ball.

If a < ^p

√r 2τ√^p

α_N =: a₁, where r is given by the geometry of the Mountain Pass Theorem,ϕ_τ ∈B_r.

On the other hand, Z

Ω

Z ϕ_r 0

H(t−a)(t⁺)^qdt dx≥ Z

{|x|≤1/2}

Z ϕ_r 0

H(t−a)(t⁺)^qdt dx

=a^q+1(τ^q+1−1) q+ 1

Z

{|x|≤1/2}

dx;

that is,

Z

Ω

Z ϕ 0

H(t−a)(t⁺)^qdt dx≥a^q+1(τ^q+1−1) q+ 1 C1

and so Iλ,a(ϕτ) = 1

pMc(kϕτk^p_1,p)−λ Z

Ω

Z ϕ_τ 0

H(t−a)(t⁺)^qdt dx− t^s+1 s+ 1

Z

Ω

h(x)(ϕ⁺_τ)^q+1. As ^t_s+1^s+1R

Ωh(x)(ϕ⁺_τ)^q+1≥0, it follows that Iλ,a(ϕτ)≥1

pMc(kϕτk^p_1,p)−λC₁a^q+1(τ^q+1−1)

q+ 1 .

BecauseM is a continuous function on [0,kϕτk^p_1,p], Iλ,a(ϕτ)≤ C2

p kϕτk^p_1,p−λC1a^q+1(τ^q+1−1) q+ 1

≤ C2

p 2^pa^pτ^pαN −λC1a^q+1(τ^q+1−1) q+ 1

=a^q+1C₂2^pa^p−(q+1)τ^pα_N

p −λC₁(τ^q+1−1) q+ 1

. We now point out that

C₂2^pa^p−(q+1)τ^pα_N

p −λC₁(τ^q+1−1)

q+ 1 ≤0⇔a^p−(q+1)≤ λC₁(τ^q+1−1)p C22^p(q+ 1)αNτ^p. Setting

a₂= 1 2

λC₁(τ^q+1−1)p C22^p(q+ 1)αNτ^p

and takinga^∗ =a^∗(λ) = min{a1, a2} it follows that infB_rIλ,a <0 fora∈(0, a^∗), which proves the claim.

By the Ekeland Variational Principle, there existsu∈Brsuch that Iλ,a(u)<inf

Br

Iλ,a+, (4.6)

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Iλ,a(u)< Iλ,a(u) +ku−uk1,p, for allu∈W₀^1,p(Ω), withu6=u. (4.7) Let us choose >0 in such a way that

0< <inf

∂B_rIλ,a−inf

B_rIλ,a

and sou∈Br.

Let γ > 0 be small enough and v ∈ W₀^1,p(Ω) with kvk1,p < 1 so that uγ = u+γv∈B_r. From (4.7) we have

Iλ,a(u)< Iλ,a(u+γv) +γkvk1,p

which implies

I_λ,a(u+γv)−I_λ,a(u) +γkvk1,p≥0.

Consequently,

−kvk1,p≤ I_λ,a(u+γv)−I_λ,a(u) γ

and so

−kvk1,p≤lim sup

γ→0

Iλ,a(u+γv)−Iλ,a(u)

γ ≤I_λ,a⁰ (u;v).

Now, since

I_λ,a⁰ (u;v) = max

µ∈∂Iλ,a(u)

hµ, vi, for allu, v∈W₀^1,p(Ω), it follows that

−kvk1,p≤I_λ,a⁰ (u;v) = max

ω∈∂Iλ,a(u)hω, vi.

Interchangingv and−vwe obtain

−kvk1,p≤ max

ω∈∂Iλ,a(u)

hω,−vi=− min

ω∈∂Iλ,a(u)

hω, vi.

Therefore,

min

ω∈∂Iλ,a(u)hω, vi ≤kvk1,p, for allv∈W₀^1,p(Ω), concluding that

sup

kvk1,p<1

min

ω∈∂Iλ,a(u)hω, vi ≤. By Fan’s Min-max theorem, we obtain

min

ω∈∂Iλ,a(u)

sup

kvk1,p<1

hω, vi ≤.

Which along with (4.6) yields the existence ofun∈Brsuch that Iλ,a(un)→ec, m(un) = min

ω∈∂I_λ,a(u_n)kωk∗→0;

that is, (un) is a Palais-Smale sequence at the levelec.

By lemma 3.4, there exists u2 ∈ W₀^1,p(Ω), where, passing to a subsequence if necessary, we obtain

u_n→u₂ inW₀^1,p(Ω), (4.8)

Iλ,a(u2) =ec= inf

B_r(0)Iλ,a<0. (4.9) Thus, u₂ is a local minimum point and consequently is a critical point of I_λ,a. Hence, following the same arguments made before, we have thatu₂is also solution

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of the problem (1.2) and using the same arguments used in (i) withu1 solution, we obtain alsou2 satisfy (i).

Proof of (ii): By (4.1) and (4.9), it follows thatIλ,a(u2)<0< Iλ,a(u1).

Proof of (iii): We will show now that|{x ∈ Ω;u_i(x) > a}| > 0, i = 1,2. We begin with the solutionu₁ obtained via the Mountain Pass Theorem. Suppose, by contradiction, thatu₁(x)≤aa.e, in Ω. So

λ Z

Ω

Z u1

0

H(t−a)(t⁺)^q = 0.

By the above equality and sinceu1is critical point ofIλ,a, we obtain [M(ku1k^p_1,p)]^p−1ku1k^p_1,p=

Z

Ω

hu^s+1₁ . Note that

Z

Ω

hu^s+1₁ ≤ |h|_∞ Z

Ω

u^p₁u^s+1−p₁ and thus

[M(ku1k^p_1,p)]^p−1ku1k^p_1,p≤ |h|_∞ Z

Ω

u^p₁u^s+1−p₁ ≤C|h|_∞a^s+1−pku1k^p_1,p which implies

[M(ku1k^p_1,p)]^p−1≤C|h|∞a^s+1−p.

Note that, there exists C > 0 such that ku1k ≥ C. Hence, using (H4); that is, M(t)>0 for allt≥0, there exists C >e 0 such that

0<Ce≤[M(ku1k^p_1,p)]^p−1≤C|h|∞a^s+1−p, for alla >0, which is impossible.

We now consider the solution u2 obtained via Ekeland Variational Principle.

Suppose, by contradiction, thatu2(x)≤aa.e. in Ω. Thus λ

Z

Ω

Z u2

0

H(t−a)(t⁺)q= 0.

By the above equality and sinceu₂is a critical point of I_λ,a, we obtain [M(ku₂k^p_1,p)]^p−1ku₂k^p_1,p=

Z

Ω

hu^s+1₂ . We will consider two cases:

Case (1): If 0<ku2k^p_1,p≤t₁, from (H₁) we have

M(ku2k^p_1,p)≥m₁>0. (4.10) Hence,

[M(ku2k^p_1,p)]^p−1ku2k^p_1,p≤ |h|∞

Z

Ω

u^p₂u^s+1−p₂ ≤C|h|∞a^s+1−pku2k^p_1,p; that is,

[M(ku₂k^p_1,p)]^p−1≤C|h|_∞a^s+1−p. So [M(ku2k^p_1,p)]→0 asa→0 which cannot happen from (4.10).

Case (2): Ifku2k^p_1,p≥t1, thenMc(ku2k^p_1,p)≥Mc(t1)>0, becauseMcis increasing.

Moreover,

0≤[M(ku2k^p_1,p)]^p−1ku2k^p_1,p= Z

Ω

hu^s+1₂ ≤ |h|_∞a^s+1.

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Thus [M(ku2k^p_1,p)]^p−1ku2k^p_1,p→0 asa→0. Since Iλ,a(u2) =Mc(ku2k^p_1,p)− 1

s+ 1 Z

Ω

hu^s+1₂ ≥Mc(t1)−[M(ku2k^p_1,p)]^p−1ku2k^p_1,p, we obtain

0<Mc(t₁)≤I_λ,a(u₂) + [M(ku₂k^p_1,p)]^p−1ku₂k^p_1,p. BecauseI_λ,a(u₂)<0, we obtain

0<Mc(t1)≤[M(ku2k^p_1,p)]^p−1ku2k^p_1,p.

Hence, asa→0, we haveMc(t1) = 0 for t1>0, which is an absurd. With this, we conclude the proof of the theorem.

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Francisco Julio S. A. Corrˆea

Universidade Federal de Campina Grande, Unidade Acadˆemica de Matem´atica e Es- tat´ıstica, CEP:58109-970, Campina Grande-PB, Brazil

E-mail address:[email protected]

Rubia G. Nascimento

Faculdade de Matemática, Universidade Federal do Pará, CEP:66075-110, Belém -PA, Brazil

E-mail address:[email protected]